Recursive construction of the operator product expansion in curved space

Fröb, Markus B.

doi:10.1007/jhep02(2021)195

Cited by 4 publications

(5 citation statements)

References 89 publications

(126 reference statements)

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“…By setting x = N f N and a = g 2 , eq. (9.21) has been conjectured in [18] to hold for QCD in the conformal window x x c close to its lower edge 12 x c = ( N f N ) c . For x = x c , eq.…”

Section: On the Merging Of Two Nontrivial Zeroesmentioning

confidence: 91%

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Low-energy theorem and OPE in the conformal window of massless QCD

Bochicchio¹,

Pallante²

2022

Preprint

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We develop a new technique, based on a low-energy theorem (LET) of NSVZ type derived in [1], for the nonperturbative investigation of SU(N) QCD with N f massless quarks -or, more generally, of massless QCD-like theories -in phases where the beta function, β(g), with g = g(µ) the renormalized gauge coupling, admits an isolated zero, g * , in the infrared (IR) or ultraviolet (UV). In the above phases, the theory is either exactly conformal invariant -if g(µ) = g * for any scale µ -or may be asymptotically conformal in the IR/UV -if g(µ) = g * for some µ = 0. In the latter case, g(µ) → g * only for µ → 0 + /µ → +∞. We point out that the LET sets constraints on 3-point correlators involving the insertion of Tr F 2 , its anomalous dimension γ F 2 , and the anomalous dimensions of multiplicatively renormalizable operators at g * . These constraints intertwine with the exact conformal scaling for g(µ) = g * and the IR/UV asymptotics -which may or may not coincide with the IR/UV limit of the aforementioned conformal scaling -for g(µ) = g * . Our new technical tool is the nonperturbative evaluation of the dimensionally regularized LETin the conformal-invariant scheme introduced in [2] -either exactly for g(µ) = g * or asymptotically in the IR/UV for g(µ) → g * . In the above cases, we also discuss how the LET for bare correlators is the rationale for the existence in massless QCD of the mysterious divergent contact term in the OPE of Tr F 2 with itself discovered in perturbation theory in [3,4] and computed to all orders in [5]. Specifically, if γ F 2 does not vanish, the divergent contact term in the rhs of the LET for the 2-point correlator of Tr F 2 has to match -and we verify by direct computation that it actually does -the divergence in the lhs due to the nontrivial anomalous dimension of Tr F 2 . Hence, remarkably, the additive renormalization due to the divergent contact term in the rhs is related by the LET to the multiplicative renormalization in the lhs, in such a way that a suitably renormalized version of the LET has no ambiguity for additive renormalization. E.3 LET for γ F 2 = 0 in d = 4 72 E.4 LET for exceptional γ F 2 = 0 in d = 4 73 E.5 LET for O = F 2 to order g 2 in perturbation theory 73

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“…By setting x = N f N and a = g 2 , eq. (9.21) has been conjectured in [18] to hold for QCD in the conformal window x x c close to its lower edge 12 x c = ( N f N ) c . For x = x c , eq.…”

Section: On the Merging Of Two Nontrivial Zeroesmentioning

confidence: 91%

“…In this respect, we should also mention the treatment of the OPE in [12] and references therein that is closely related to the LET, but actually independent of the approach in [1,11] that is pursued in the present paper.…”

Section: Introduction and Physics Motivationsmentioning

confidence: 99%

Low-energy theorem and OPE in the conformal window of massless QCD

Bochicchio¹,

Pallante²

2022

Preprint

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“…Then, as shown in [22,Lemma 3.1], there is a canonical injective unital * -homomorphism α ψ : W(M, g ab ) → W(M ′ , g ′ ab ). We demand that the definition of 14 These axioms differ from the ones originally given in [22] in that the Leibniz rule W4 and the conservation of stress-energy W8 have been added as in [29]. In addition, the analytic dependence condition of [22,29] has been replaced by the joint smoothness condition of [30,31].…”

Section: It Is Useful To Viewmentioning

confidence: 99%

“…The OPE flow relations (1.2) and their generalization to other interacting theories apply for the case of flat Euclidean space. Recently, Fröb [14] has generalized these relations to quantum fields on curved Riemannian spaces, without, however, imposing the condition that the OPE coefficients be locally and covariantly defined. Since the physical world is Lorentzian, it would be of interest to generalize the flow relations to Lorentzian spacetimes.…”

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confidence: 99%

“…As established in Proposition 10, this identity implies there exists a ≡ a B T 0 {V C} such that:Q(Λ) = (d 0 a)(Λ) = (D(Λ) − I)a. (E.13)For tensor-valued43 Q, the results of Appendix C imply the a can be inductively constructed (modulo Lorentz-invariant tensors) from j −L κρ B κρ + 4 i<j≤n η µ i µ j tr ij a , (E 14). withB κρ ≡ (B κρ ) E T 0 {V C} = 2i d D yy [κ ∂ ρ]χ(y, 0)C E T 0 {V C} (y, 0; 0).…”

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Local and Covariant Flow Relations for OPE Coefficients in Lorentzian Spacetimes

Klehfoth¹,

Wald²

2022

Preprint

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For Euclidean-signature quantum field theories with renormalizable self-interactions, Holland and Hollands have shown the operator product expansion (OPE) coefficients satisfy "flow equations": For a (renormalized) self-coupling parameter λ, the partial derivative of any OPE coefficient with respect to λ is given by an integral over Euclidean space of a sum of products of other OPE coefficients evaluated at λ. These Euclidean flow equations were proven to hold order-by-order in perturbation theory, but they are well defined non-perturbatively and thus provide a possible route towards giving a non-perturbative construction of the interacting field theory. The purpose of this paper is to generalize the Holland and Hollands results for flat Euclidean space to curved Lorentzian spacetimes in the context of the solvable "toy model" of massive Klein-Gordon scalar field theory on globally-hyperbolic curved spacetimes, with the squared mass, m 2 , viewed as the "self-interaction parameter". There are a number of difficulties that must be overcome to carry out this program. Even in Minkowski spacetime, a serious difficulty arises from the fact that all integrals must be done over a compact region of spacetime to ensure convergence. However, there does not exist any Lorentz-invariant function of compact support, so any flow relations that involve only integration over a compact region cannot be Lorentz covariant. We show how covariant flow relations can be obtained by the addition of "counterterms" that cancel the non-covariant dependence on the cutoff function in a manner similar to that used in the Epstein-Glaser renormalization scheme. The necessity of integration over a finite region also effectively introduces an "infrared cutoff scale" L into the flow relations, *

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Low-energy theorem revisited and OPE in massless QCD

Bochicchio,

Pallante

2024

J. High Energ. Phys.

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We revisit a low-energy theorem (LET) of NSVZ type in SU(N) QCD with Nf massless quarks derived in [1] by implementing it in dimensional regularization. The LET relates n-point correlators in the l.h.s. to n + 1-point correlators with the extra insertion of TrF2 at zero momentum in the r.h.s. We demonstrate that, for 2-point correlators of an operator O in the l.h.s., the LET implies that, in general, the integrated 3-point correlator in the r.h.s. needs in perturbation theory an infinite additive renormalization in addition to the multiplicative one. We relate the above counterterm to a corresponding divergent contact term in a certain coefficient of the OPE of TrF2 with O in the momentum representation, thus extending to any operator O an independent argument that first appeared for O = TrF2 in [2]. Finally, we demonstrate that in the asymptotically free phase of QCD the aforementioned counterterm in the LET is actually finite nonperturbatively after resummation to all perturbative orders. We also briefly recall the implications of the LET in the gauge-invariant framework of dimensional regularization for the perturbative and nonperturbative renormalization in large-N QCD. The implications of the LET inside and above the conformal window of SU(N) QCD with Nf massless quarks will appear in a forthcoming paper.

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Recursive construction of the operator product expansion in curved space

Cited by 4 publications

References 89 publications

Low-energy theorem and OPE in the conformal window of massless QCD

Low-energy theorem and OPE in the conformal window of massless QCD

Local and Covariant Flow Relations for OPE Coefficients in Lorentzian Spacetimes

Low-energy theorem revisited and OPE in massless QCD

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